Moment of inertia of triangular plate about its centroid
Answers
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eight of triangle is h = ¼ L√3
if triangle is one of its edge on below position, so the height of center is ⅔ h.
I = ∫ r² dA
x = yL/h =
dA = x dy = (yL/h) dy
moment inertia with respect to X-axis is
Ix = ∫ y² dA = ∫ y² (yL/h) dy = (L/h) ∫ y³ dy
Ix = (L/h) (¼) y⁴ which y from 0 to h
Ix = (L/h) (h⁴ - 0) = ¼ Lh³
according to parallel axis theorem, calculate moment inertia w.r.t X-axis through the center of mass is,
Ix = Ixo + Ay²
¼ Lh³ = Ixo + (½ Lh) (⅔h)²
Ixo = (1/36) Lh³
Let's we calculate moment inertia with respect to Y-axis through the center of mass is,
Iy = ∫ x² dA
dA = y dx
y = -(x - h)
Iyo = ∫ x² dA = 2 ∫ x² (-(x - h)) dx from x = 0 to x = ½ L
Iyo = 2 ∫ (-x³ + x²h) dx = -½ x⁴ + ⅔ x³h from x = 0 to x = ½ L
Iyo = (-1/32) L⁴ + (1/12) L³h
notify that Y-axis through the center of mass.
according to Pythagoras theorem,
r² = x² + y²
∫ r² dA = ∫ x² dA + ∫ y² dA
Ir = Iyo + Ixo
Ir = (-1/32) L⁴ + (1/12) L³h + (1/36) Lh³
but h = ¼ L√3
Ir = L⁴(17√3 - 24)/768
moment of inertia of an equilateral triangular plate of length L about its centre is Ir = L⁴(17√3 - 24)/768
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Answer:
Let the mass density be ρkg/m
2
Let the side PQ=RS=a & QR=PS=b (From fig a>b)
Mass of rectangle is m= ρab
Therefore Moment of Inertia of Rectangle about its center = m
12
a
2
+b
2
Distance of P point from center of rectangle is
2
a
2
+b
2
Therefore Moment of Inertia of Rectangle about P, I= m
12
a
2
+b
2
+m
4
a
2
+b
2
=m
3
a
2
+b
2
Mass of triangle PQR=
2
m
=
2
ρab
Moment of Inertia of Triangle PQR about its centroid = ρ
12
ab
3
+ba
3
=m
12
a
2
+b
2
Distance of point P from centroid =
(
3
2a
)
2
+(
3
b
)
2
Moment of Inertia of Triangle PQR about P= m
12
a
2
+b
2
+
2
m
{(
3
2a
)
2
+(
3
b
)
2
}=
2
m
18
11a
2
+5b
2
>
2
m
18
6a
2
+6b
2
>
2
I
(As a>b)
Distance of point R from centroid =
(
3
2b
)
2
+(
3
a
)
2
Moment of Inertia of TrianglePQR about P= m
12
a
2
+b
2
+
2
m
{(
3
a
)
2
+(
3
2b
)
2
}=
2
m
18
5a
2
+11b
2
As a>b therefore
2
m
18
5a
2
+11b
2
can be less than or equal to or greater than
2
I